Context. Aims. Super magnetized neutron stars of ∼1015 G, magnetars, and magnetized protoneutron stars born after the magnetically-driven supernovae are likely to have very strong toroidal magnetic fields. Methods. Long-term, three-dimensional general relativistic magnetohydrodynamic simulations were performed to prepare isentropic neutron stars with toroidal magnetic fields in equilibrium as initial conditions. To explore the effects of rotations on the stability, simulations were done for both nonrotating and rigidly rotating models. Results. We find the emergence of the Parker and/or Tayler instabilities in both the nonrotating and rotating models. For both nonrotating and rotating models, the Parker instability is the primary instability predicted by the local linear perturbation analysis. The interchange instability also appears in the rotating models. It is found that the Parker instability cannot be suppressed even if the stars rotate rapidly. This finding does not agree with the perturbation analysis, because rigidly and rapidly rotating stars are marginally stable; therefore, in the presence of stellar pulsations that deform the rotational profile, unstable regions develop with a negative gradient of the angular momentum profile. After the onset of the instabilities, a turbulence is excited. In contrast to the axisymmetric case, the magnetic fields never reach a state of equilibrium after the the turbulence develops. Conclusions. Isentropic neutron stars with strong toroidal magnetic fields are always likely to be unstable against the Parker instability. Turbulent motion is induced and maintained for a long time. This conclusion is different for axisymmetric simulations and suggests that three-dimensional simulation is indispensable for exploring the formation of magnetars or the prominent activities of magnetars such as giant flares.
- Stars: magnetars
- Stars: neutron